Interacting Fisher–Wright diffusions in a catalytic medium

We study the longtime behaviour of interacting systems in a randomly fluctuating (space–time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher–Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium). Here we introduce a model of interacting Fisher–Wright diffusions where the local resampling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d-dimensional lattice. While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium. Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001     

Inégalités de concentration pour les processus empiriques de classes de parties

We propose new concentration inequalities for maxima of set-indexed empirical processes. Our approach is based either on entropy inequalities or on martingale methods. The improvements we get concern the rate function which is exactly the large deviations rate function of a binomial law in most of the cases. Received: 11 January 2000 / Revised version: 12 May 2000 / Published online: 14 December 2000     

Aging of spherical spin glasses

Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for (hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging” phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures. Received: 8 July 1999 / Revised version: 2 June 2000 / Published online: 6 April 2001     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001     

The high temperature case for the random K-sat problem

We give a completely rigorous proof that the replica-symmetric solution holds at high enough temperature for the random K-sat problem. The most notable feature of this problem is that the order parameter of the system is a function and not a number. Received: 21 April 1998 / Revised version: 24 April 2000 / Published online: 21 December 2000     

Rescaled contact processes converge to super-Brownian motion in two or more dimensions

We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics. Received: 2 February 1998 / Revised version: 28 August 1998     

Homogenization of divergence-form operators with lower-order terms in random media

The probabilistic machinery (Central Limit Theorem, Feynman-Kac formula and Girsanov Theorem) is used to study the homogenization property for PDE with second-order partial differential operator in divergence-form whose coefficients are stationary, ergodic random fields. Furthermore, we use the theory of Dirichlet forms, so that the only conditions required on the coefficients are non-degeneracy and boundedness. Received: 27 August 1999 / Revised version: 27 October 2000 / Published online: 26 April 2001     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

Stopping games with randomized strategies

We study stopping games in the setup of Neveu. We prove the existence of a uniform value (in a sense defined below), by allowing the players to use randomized strategies. In constrast with previous work, we make no comparison assumption on the payoff processes. Moreover, we prove that the value is the limit of discounted values, and we construct ε-optimal strategies. Received: 10 May 1999 / Revised version: 18 May 2000 / Published online: 15 February 2001     

Infinite-dimensional Langevin equations: uniqueness and rate of convergence for finite-dimensional approximations

The paper deals with the infinite-dimensional stochastic equation dX= B(t, X) dt + dW driven by a Wiener process which may also cover stochastic partial differential equations. We study a certain finite dimensional approximation of B(t, X) and give a qualitative bound for its rate of convergence to be high enough to ensure the weak uniqueness for solutions of our equation. Examples are given demonstrating the force of the new condition. Received: 6 November 1999 / Revised version: 21 August 2000 / Published online: 6 April 2001     

Random walk on upper triangular matrices mixes rapidly

We present an upper bound O(n ² ) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant, and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process. Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000     

A non-uniform Berry–Esseen bound via Stein's method

This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments. Received: 2 March 2000 / Revised version: 20 July 2000 / Published online: 26 April 2001     

Slowdown and neutral pockets for a random walk in random environment

We consider a d-dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift “pointing to the right”. We obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion, both under the annealed and quenched measures. These estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk. Received: 12 March 1998 / Revised version: 19 February 1999     

Metastability in stochastic dynamics of disordered mean-field models

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distributions of the rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model. Received: 26 November 1998 / Revised version: 21 March 2000 / Published online: 14 December 2000     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

Large deviations for small noise diffusions with discontinuous statistics

This paper proves the large deviation principle for a class of non-degenerate small noise diffusions with discontinuous drift and with state-dependent diffusion matrix. The proof is based on a variational representation for functionals of strong solutions of stochastic differential equations and on weak convergence methods. Received: 26 May 1998 / Revised version: 24 February 1999